As it well known, Bell numbers denoted $B_{n}$ counts distinct partitions of the finite set $[n]$. So for example if $n=3$ there are 5 ways to the set $\left\{ a,b,c\right\}$ can be partitioned:

$$\left\{ \left\{a\right\}, \left\{b\right\}, \left\{c\right\} \right\}$$ $$\left\{ \left\{a\right\}, \left\{b,c\right\} \right\}$$ $$\left\{ \left\{b\right\}, \left\{a,c\right\} \right\}$$ $$\left\{ \left\{c\right\}, \left\{a,b\right\} \right\}$$ $$\left\{ \left\{a,b,c\right\} \right\}$$

My question is what is known (or is it unknown?) about the cardinality of odd(or even) number of parts in partitions of Bell numbers.

So for the above example, the first partition and the last partition is with odd number of parts and the rest are with even number of parts. In which case we have $B^e_{3}=3$ and $B^o_{3}=2$